nag_sum_fft_qtrsine (c06rgc) (PDF version)
c06 Chapter Contents
c06 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_sum_fft_qtrsine (c06rgc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_sum_fft_qtrsine (c06rgc) computes the discrete quarter-wave Fourier sine transforms of m sequences of real data values. The elements of each sequence and its transform are stored contiguously.

2  Specification

#include <nag.h>
#include <nagc06.h>
void  nag_sum_fft_qtrsine (Nag_TransformDirection direct, Integer m, Integer n, double x[], NagError *fail)

3  Description

Given m sequences of n real data values xjp , for j=1,2,,n and p=1,2,,m, nag_sum_fft_qtrsine (c06rgc) simultaneously calculates the quarter-wave Fourier sine transforms of all the sequences defined by
x^ k p = 1n j=1 n-1 xjp × sin j 2k-1 π2n + 12 -1 k-1 xnp ,   if ​ direct=Nag_ForwardTransform ,
or its inverse
xkp = 2n j=1 n x^ j p × sin 2j-1 k π2n ,   if ​ direct=Nag_BackwardTransform ,
where k=1,2,,n and p=1,2,,m.
(Note the scale factor 1n  in this definition.)
A call of nag_sum_fft_qtrsine (c06rgc) with direct=Nag_ForwardTransform followed by a call with direct=Nag_BackwardTransform will restore the original data.
The two transforms are also known as type-III DST and type-II DST, respectively.
The transform calculated by this function can be used to solve Poisson's equation when the solution is specified at the left boundary, and the derivative of the solution is specified at the right boundary (see Swarztrauber (1977)).
The function uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, described in Temperton (1983), together with pre- and post-processing stages described in Swarztrauber (1982). Special coding is provided for the factors 2, 3, 4 and 5.

4  References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Swarztrauber P N (1977) The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle SIAM Rev. 19(3) 490–501
Swarztrauber P N (1982) Vectorizing the FFT's Parallel Computation (ed G Rodrique) 51–83 Academic Press
Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys. 52 340–350

5  Arguments

1:     directNag_TransformDirectionInput
On entry: indicates the transform, as defined in Section 3, to be computed.
Forward transform.
Inverse transform.
Constraint: direct=Nag_ForwardTransform or Nag_BackwardTransform.
2:     mIntegerInput
On entry: m, the number of sequences to be transformed.
Constraint: m1.
3:     nIntegerInput
On entry: n, the number of real values in each sequence.
Constraint: n1.
4:     x[n×m]doubleInput/Output
On entry: the m data sequences to be transformed. The pth sequence to be transformed, denoted by xjp, for j=1,2,,n and p=1,2,,m, must be stored in x[p-1×n+j-1].
On exit: the m quarter-wave sine transforms, overwriting the corresponding original sequences. The n components of the pth quarter-wave sine transform, denoted by x^kp, for k=1,2,,n and p=1,2,,m, are stored in x[p-1×n+k-1].
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
On entry, m=value.
Constraint: m1.
On entry, n=value.
Constraint: n1.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

7  Accuracy

Some indication of accuracy can be obtained by performing a subsequent inverse transform and comparing the results with the original sequence (in exact arithmetic they would be identical).

8  Parallelism and Performance

nag_sum_fft_qtrsine (c06rgc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The time taken by nag_sum_fft_qtrsine (c06rgc) is approximately proportional to nm logn, but also depends on the factors of n. nag_sum_fft_qtrsine (c06rgc) is fastest if the only prime factors of n are 2, 3 and 5, and is particularly slow if n is a large prime, or has large prime factors. Workspace is internally allocated by this function. The total amount of memory allocated is On double values.

10  Example

This example reads in sequences of real data values and prints their quarter-wave sine transforms as computed by nag_sum_fft_qtrsine (c06rgc) with direct=Nag_ForwardTransform. It then calls the function again with direct=Nag_BackwardTransform and prints the results which may be compared with the original data.

10.1  Program Text

Program Text (c06rgce.c)

10.2  Program Data

Program Data (c06rgce.d)

10.3  Program Results

Program Results (c06rgce.r)

nag_sum_fft_qtrsine (c06rgc) (PDF version)
c06 Chapter Contents
c06 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014